Magic constant

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The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. In general where is the side length of the square.



For normal magic squares of orders n = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260 (sequence A006003 in the OEIS). For example, a normal 8x8 square will always equate to 260 for each row, column, or diagonal.

The term magic constant or magic sum is similarly applied to other "magic" figures such as magic stars and magic cubes. Number shapes on a triangular grid divided into equal polyiamond areas containing equal sums give polyiamond magic constant. [1]

Magic stars

The magic constant of an n-pointed normal magic star is .

Magic series

In 2013 Dirk Kinnaes found the magic series polytope. The number of unique sequences that form the magic constant is now known up to . [2]


In the mass model, the value in each cell specifies the mass for that cell. [3] This model has two notable properties. First it demonstrates the balanced nature of all magic squares. If such a model is suspended from the central cell the structure balances. ( consider the magic sums of the rows/columns .. equal mass at an equal distance balance). The second property that can be calculated is the moment of inertia. Summing the individual moments of inertia ( distance squared from the center x the cell value) gives the moment of inertia for the magic square. "This is the only property of magic squares, aside from the line sums, which is solely dependent on the order of the square." [4]

See also

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In mathematics, a perfect magic cube is a magic cube in which not only the columns, rows, pillars, and main space diagonals, but also the cross section diagonals sum up to the cube's magic constant.

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Most-perfect magic square

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  1. Each 2×2 subsquare, including wrap-round, sums to s/k, where s = n(n2 + 1)/2 is the magic sum.
  2. All pairs of integers distant n/2 along any diagonal are complementary.

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(2,1)-Pascal triangle

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  2. Walter Trump
  3. Heinz 3 dimensional magic square/
  4. Peterson